Pure torsion for incompressible hyperelastic materials of Valanis–Landel form

Abstract

The stretch-based formulation of basic problems for incompressible isotropic hyperelastic materials has been the subject of renewed recent attention largely motivated by application to modelling the mechanical response of soft tissues. Here, we are concerned with this formulation of the classic problem of simple torsion of a circular cylinder that has been primarily in the past investigated by employing the classical approach of Rivlin in terms of invariants of the Cauchy–Green strain tensor. Attention is focused on examining the resultant axial force necessary to sustain pure torsion, thereby investigating the Poynting effect, and the resultant moment needed to generate the twist. General results for the resultant force and moment are obtained for all hyperelastic materials that can be modelled in the celebrated separable Valanis–Landel form. These expressions are specialized for the one-term Ogden model to provide explicit results valid for all strain-stiffening exponents in this model. Some particular examples are provided, mainly for even values of the exponent. A notable result obtained for the Ogden model is that for values of the hardening exponent n in that model for which n ≤ 6 , one obtains the classical Poynting effect, whereas for n > 6 , a reverse Poynting effect always occurs for sufficiently small twist.